MATC Mathematics Club
Problem Set Volume 2 Page

The Puzzle Corner of the MATC Mathematics Club
Madison Area Technical College
Madison, Wisconsin

Volume 2, Spring 2002 Problem Sets
Problem Set #5 ( Click here for the solutions ).
1. Let A be the sum of ten positive real numbers and let B be the sum of the reciprocals of these ten numbers.  What is the smallest possible value for AB?
2. The sum of the lengths of the diagonals of a rectangle is 8 square roots of thirteen, and its area is 24.  What is the perimeter of the rectangle?
3. Determine how many 2 digit numbers satisfy the following property:  When the number is added to the number obtained by reversing the digits, the sum is 132.
Problem Set #4 ( Click here for the solutions ).
1. You have three boxes, one containing grapes, one containing olives, and one containing grapes and olives.  The boxes were labeled for their content, but someone has switched the labels so that every box is now incorrectly labeled.  You are allowed to take one item at a time out of any box, without looking inside, and by this process of sampling you are to determine the contents of all three boxes.  What is the smallest number of drawings needed to do this?
2. Given that a match has a length of 1 unit, it is possible to place 12 matches on a plane in various ways to form polygons with integral areas. For example, one could construct a square with an area of 9 sq. units and a cross with an area of 5 sq. units.  Can you use all 12 matches to form the perimeter of a polygon with an area of exactly 4 sq. units?  The entire length of each match must be used.
3. Two circles of equal radius intersect each other such that each circle passes through the center of the other circle. Find the perimeter of this figure.
Problem Set #3 ( Click here for the solutions ).
1. Imagine that the home of your math instructor is located at the origin of a Cartesian coordinate system and MATC is located at (4,4).  To get to MATC, your instructor always walks either North or East.  How many different routes can your instructor take to work?
2. Mr. A, Ms. B and Mrs. C all apply for the same job.  Each is qualified, so the employer gives the following test. The applicants are blindfolded, then lined up facing the employer. Ms. B in front of him, Mr. A behind her, and Mrs. C behind Mr. A.  The employer then says, "I have 3 blue hats and 2 gray hats.  I am going to select 3 hats and place one on each of your heads.  Iāll then remove the blindfold and ask you to identify the color of the hat on your head.  The first person to identify the color of his/her hat will get the job.  If you guess wrong, you will be disqualified." The blindfolds were removed and the employer said, "Mrs. C, you begin, to be followed by Mr. A then Mrs. B."  Mrs. C who could see the other two applicants said she could not determine the color of her hat.  Mr. A  who could only see Ms. B said the same.  But Ms., B who could see neither, correctly named the color of her hat and won the job.  What color hat was she wearing and how did she know?  Explain.
3. ABCD is a square of side 8 and Y is the point of intersection of the diagonals of the square.  XYZ is a right triangle with right angle at Y and P, Q are points of intersections of sides YX and YZ with sides BC, CD respectively.  If BP is 2, whatās the area of the overlap CPYQ?
Problem Set #2 ( Click here for the solutions ).
1. Suppose there is a 3X3 grid with a 3/8 in the upper left cell and a ¼ in the second row third column.  Determine the remaining cells so that the row sum, column sum and the diagonal sum are equal 1.
2. In the array of numbers given below, determine the column the row and column in which the number 1000 appears.


  2 3 4 5  
9 8 7 6    
  10 11 12 13  
17 16 15 14    
  18 19 20 21  
25 24 23 22    


3. Write the next number in the series {1, 11, 21, 1211, 111221,·.}
Problem Set #1 ( Click here for the solutions ).
1. Each of the following numbers, 1324, 1342, 2314, 3124, 4132 and 4231 represents a word in the English Language.  If each digit represents the same letter in all of them, what are the words?
2. What do the following series represent?
  1. 4,8,12,2,1,7,6,3,5,11,10,9
  2. 7,8,5,5,3,4,4,6,9,7,8,8
  3. 3,3,5,4,4,3,5,5,4,3,·.
3. A man is jogging across a bridge.  When heās 3/8 ths of the way across, he hears a train coming from behind him.  He calculates that if he keeps running, he will reach the end of the bridge at the same instant as the train.  He also calculates that if he turns around and runs back, he will reach the beginning of the bridge at the same instant as the train.  Say the man runs consistently at 8 mph, what is the speed of the train?



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